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Nx--B u fy 2 44 - - - - -

②§#,,§§fc×,y, dxdy = of x. yydydx

I2

Example 10. Find the volume of the solid bounded by the cylinder x2 + y2 = 4 and the planesy + z = 4 and z = 0.

From: Calculus Early Transcendentals,10th edition, Howard Anton, Irl C. Beven,Stephen Deavis, page 1013

Reversing the Order of Integration

Sometimes the evaluation of an iterated integral can be simplified by reversing the order ofintegration. The next example illustrates how this is done.

Example 11. Since there is no elementary antiderivative of ex2, the integral

∫ 2

0

∫ 1

y/2ex

2dx dy

cannot be evaluated by performing the x-integration rst. Evaluate this integral by express-ing it asan equivalent iterated integral with the order of integration reversed.

Solution.

10

* feidx=

x--

X -- Iz ⇒ y -- 2x1 ' 2x

^

⇐µ""

9. fue"dxdy= f f E''

dydx

R--§ye×2/yi×d×#g

X- I =§2xe×' dxhi u- X

'

; du -- 2x DX

VOI : X = o Toi u = O

XI 195 U = 1

I!§e" du -- e'- E -

- e - z #

Area calculated as a double integral

area of R =

∫∫

R1 dA =

∫∫

RdA

From: Calculus Early Transcendentals, 10th edition, Howard Anton, Irl C. Beven, StephenDeavis, page 1014

11

In w.n.br :env Xy ban ffrfcx, yidAset fix

, y ) -. I ⇐ ffrfcx, and A = Sfpd A

.

'

. ffr DA do win. arouses R

Example 12. Use a double integral to nd the area of the region R enclosed between the parabola

y =x2

2and the line y = 2x

From: Calculus Early Transcendentals,10th edition, Howard Anton, Irl C. Beven,Stephen Deavis, page 1014

EXERCISES 3.1.2

1-8 Evaluate the iterated integrals.

1.

∫ 1

0

∫ x

x2xy2 dy dx

2.

∫ 3/2

1

∫ 3−y

yy dx dy

3.

∫ 3

0

∫ √9−y2

0y dx dy

4.

∫ 1

1/4

∫ x

x2

√x

ydy dx

5.

∫ √2π

√π

∫ x2

0sin

y

xdy dx

6.

∫ 1

−1

∫ x2

−x2(x2 − y) dy dx

7.

∫ 1

0

∫ x

0y√

x2 − y2 dy dx

8.

∫ 2

1

∫ y2

0ex/y

2dx dy

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Midterm F.21 Feb . 15.30-18.30

-

12135105